Edge states for the Kalmeyer-Laughlin wave function
Benedikt Herwerth, Germán Sierra, Hong-Hao Tu, J. Ignacio Cirac, Anne E. B. Nielsen
EEdge states for the Kalmeyer-Laughlin wave function
Benedikt Herwerth, ∗ Germ´an Sierra, Hong-Hao Tu, J. Ignacio Cirac, and Anne E. B. Nielsen
1, 3 Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Instituto de F´ısica Te´orica, UAM-CSIC, Madrid, Spain Department of Physics and Astronomy, Aarhus University,Ny Munkegade 120, DK-8000 Aarhus C, Denmark
We study lattice wave functions obtained from the SU(2) Wess-Zumino-Witten conformal fieldtheory. Following Moore and Read’s construction, the Kalmeyer-Laughlin fractional quantum Hallstate is defined as a correlation function of primary fields. By an additional insertion of Kac-Moodycurrents, we associate a wave function to each state of the conformal field theory. These wavefunctions span the complete Hilbert space of the lattice system. On the cylinder, we study globalproperties of the lattice states analytically and correlation functions numerically using a MetropolisMonte Carlo method. By comparing short-range bulk correlations, numerical evidence is providedthat the states with one current operator represent edge states in the thermodynamic limit. Weshow that the edge states with one Kac-Moody current of lowest order have a good overlap with low-energy excited states of a local Hamiltonian, for which the Kalmeyer-Laughlin state approximatesthe ground state. For some states, exact parent Hamiltonians are derived on the cylinder. TheseHamiltonians are SU(2) invariant and nonlocal with up to four-body interactions.
PACS numbers: 71.27.+a, 73.43.-f, 11.25.Hf
I. INTRODUCTION
The discovery of the fractional quantum Hall (FQH)effect has revealed the physical existence of a new,strongly correlated state of matter. One surprising prop-erty of FQH phases is the fact that they cannot be clas-sified in terms of symmetries . This is a fundamentaldifference to most other known states of matter and ledto the concept of classifying states in terms of topologicalorder . Topological phases are not characterized by alocal order parameter but their properties depend on thetopology of the system. For example, the ground statedegeneracy of a FQH system was shown to depend onthe genus of the surface on which the state is defined .A central characteristic of topological order is the ap-pearance of gapless edge states. Such edge modes areknown to be present in a FQH system even though thebulk is gapped. It was shown that these edge states canbe used to characterize the topological order of a FQHstate . Furthermore, they are of particular physical im-portance because they determine the transport proper-ties of the system. The strongly correlated state formedby the edge excitations of a FQH system is that of a chi-ral Luttinger liquid and its theoretical properties are re-lated to Kac-Moody algebras . For continuous systems,edge state wave functions were constructed as correlationfunctions of a conformal field theory (CFT) and interms of Jack polynomials .The use of 1 + 1 dimensional CFT in describing FQHstates was pioneered by Moore and Read . In thisapproach, a trial wave function for the FQH state isgiven as the correlation function of conformal primaryfields. Laughlin’s wave function for the continuum as well as its SU(2) symmetric, bosonic lattice version,the Kalmeyer-Laughlin (KL) state , were shown tobe of this form . It was already conjectured by Moore and Read that the FQH edge modes shouldbe described by the same CFT that defines the bulkwave function. The idea of this bulk-edge correspon-dence was later developed into an extension of Moore andRead’s method . Within this approach, trial FQHedge states are obtained from CFT descendant states.This paper is based on a similar ansatz for edge statesin lattice systems. Our starting point is the SU(2) Wess-Zumino-Witten (WZW) theory, which was usedpreviously to obtain the KL state as a correlationfunction of primary fields. By insertion of Kac-Moodycurrents, we define a tower of states, which correspondsto the CFT descendant states. Since we work in a spinformulation, the Hilbert space of the lattice system isthat of N spin- degrees of freedom, where occupied sitesare represented as spin-up and empty sites as spin-down.We show that the mapping from CFT states to spinstates is surjective, i.e. any state of the spin system canbe written as a linear combination of states constructedfrom the CFT. As a consequence, not all states obtainedin this way are edge states. For some of the wave func-tions, we carry out numerical calculations to test if theydescribe edge states: On the cylinder, we compare spincorrelation functions in the states with one current op-erator to the KL state using a Metropolis Monte Carloalgorithm . We show that the nearest-neighbor bulkcorrelations approach each other exponentially as thethermodynamic limit is taken. This indicates that thestates with one current operator indeed describe edgestates.In the past years, parent Hamiltonians of the KLstate and its non-Abelian generalizations were constructed. It was also shown that the KL state and non-Abelian FQH lattice states have a goodoverlap with the ground states of certain local Hamil-tonians. For one of these Hamiltonians, an implemen- a r X i v : . [ c ond - m a t . s t r- e l ] J a n tation scheme in ultracold atoms in optical lattices wasproposed . In this work, we study the local Hamilto-nians of Ref. 30 on the cylinder and use the KL stateand the edge states with one current operator of orderone as ansatz states. We find that there is a choice ofparameters for which both the ground state and somelow-energy excited states have a good overlap with ouransatz states.Furthermore, we construct Hamiltonians for which theKL state and some of the states obtained from it by in-sertion of current operators are exact ground states. Theproposed parent Hamiltonians are SU(2) invariant andnonlocal with up to four-body interactions.This paper is structured as follows: In Sec. II, we intro-duce the CFT model, define the map to a spin- system,and show that it is surjective. In Sec. III, properties ofour states on the cylinder are studied and numerical ev-idence is given that the states with one current operatordescribe edge states. We consider a local model Hamil-tonian in Sec. IV and derive exact parent Hamiltoniansfor some states on the cylinder in Appendix B. The con-clusion is given in Sec. V. II. CFT MODEL AND SPIN- STATES
In this section, we review some properties of the SU(2) Wess-Zumino-Witten (WZW) theory and define the cor-respondence between states of the CFT and states of aspin- system on the lattice. It is shown that this mapof CFT states to lattice spin states is surjective. A. SU(2) Wess-Zumino-Witten theory
We consider the chiral sector of the SU(2) WZW the-ory. In addition to conformal invariance, this theoryexhibits an SU(2) symmetry generated by the currents J a ( z ) with a ∈ { x, y, z } . The modes J an are defined interms of the Laurent expansion J a ( z ) = ∞ (cid:88) n = −∞ J an z − n − ,J an = (cid:73) d z πi z n J a ( z ) , (1)and satisfy the Kac-Moody algebra (cid:2) J am , J bn (cid:3) = iε abc J cm + n + m δ ab δ m + n, . (2)Here, ε abc is the Levi-Civita symbol and δ ab the Kro-necker delta. For indices c ∈ { x, y, z } , we adopt the con-vention that indices occurring twice are summed over,unless explicitly stated otherwise.The SU(2) WZW theory has two primary fields,the identity with conformal weight h = 0, and a two-component field φ s ( z ) ( s = ±
1) with conformal weight h = . The field φ s ( z ) provides a spin- irreduciblerepresentation through its operator product expansion(OPE) with the SU(2) currents : J a ( z ) φ s ( w ) ∼ − z − w (cid:88) s (cid:48) = ± t ass (cid:48) φ s (cid:48) ( w ) , (3)where t a = σ a are the SU(2) spin operators.The field content of the SU(2) WZW theory can berepresented in terms of the chiral part ϕ ( z ) of a free bosonfield as φ s ( z ) = e iπ ( q − s +1) / : e isϕ ( z ) / √ : ,J z ( z ) = − i √ ∂ z φ ( z ) ,J ± ( z ) = J x ( z ) ± iJ y ( z ) = e iπ ( q − : e ∓ i √ ϕ ( z ) : . (4)Here, q = 0 if the operators act on a state with an evennumber of modes of the h = primary field and q = 1otherwise, and the colons denote normal ordering. Thevalue of s ∈ {− , } equals two times the spin- z eigen-value of φ s ( z ).A general state in the identity sector of the CFTHilbert space is a linear combination of states( J a l − n l . . . J a − n )(0) | (cid:105) , (5)where | (cid:105) is the CFT vacuum and n i are positive integers.The sum k = (cid:80) li =1 n i defines the level of the state. Bymeans of the Kac-Moody algebra of Eq. (2), a basis canbe chosen for which the mode numbers are ordered: n l ≥ n l − ≥ · · · ≥ n > B. Spin states on the lattice
To each CFT state ( J a l − n l . . . J a − n )(0) | (cid:105) , we associatea state | ψ a l ...a n l ...n (cid:105) = (cid:88) s ,...,s N ψ a l ...a n l ...n ( s , . . . , s N ) | s , . . . , s N (cid:105) (6)in the Hilbert space of a system of N spin- degrees offreedom. Its spin wave function is defined as the CFTcorrelator ψ a l ...a n l ...n ( s , . . . , s N )= (cid:104) φ s ( z ) . . . φ s N ( z N )( J a l − n l . . . J a − n )(0) (cid:105) , (7)where (cid:104) . . . (cid:105) denotes the expectation value of radially or-dered operators in the CFT vacuum.In the sum of Eq. (6), s i = ± | s , . . . , s N (cid:105) isthe tensor product of eigenstates | s i (cid:105) of the spin op-erator t zi at position i ( t zi | s i (cid:105) = s i | s i (cid:105) ). The com-plex coordinates z i are parameters of the wave function ψ a l ...a n l ...n ( s , . . . , s N ) and define a lattice of positions inthe complex plane. Since we want to keep them fixed,we do not explicitly indicate the parametric dependenceof ψ a l ...a n l ...n ( s , . . . , s N ) on the positions z i for simplicity ofnotation.The wave function corresponding to the CFT vacuumis given by ψ ( s , . . . , s N ) ≡ (cid:104) φ s ( z ) . . . φ s N ( z n ) (cid:105) (8)= δ s χ s N (cid:89) i One may ask if the linear transformation that mapsCFT states ( J a l − n l . . . J a − n )(0) | (cid:105) to spin states ψ a l ...a n l ...n issurjective, i.e. whether any state in the Hilbert space H N of N spin- particles can be written as a linear com-bination of the states ψ a l ...a n l ...n . We now show that this isindeed the case. Introducing the N × N matrix Z = . . . z ) − . . . ( z N ) − ( z ) − . . . ( z N ) − ... ... ...( z ) − ( N − . . . ( z N ) − ( N − , (15)the definition of Eq. (11) becomes u a u a − ... u a − ( N − = Z t a t a ... t aN . (16)The determinant of Z is the well known Vandermondedeterminant: det ( Z ) = N (cid:89) i In this section, we study properties of the states definedin Sec. II B on the cylinder.We consider a square lattice with N x lattice sites inthe open direction and N y lattice sites in the periodicaldirection of the cylinder. After mapping the cylinder tothe complex plane, the coordinates assume the form z j = e πNy ( j x + ij y ) e − πNy Nx +12 . (19)Here, j x ∈ { , . . . , N x } is the x -component of the indexand j y ∈ { , . . . , N y } is the y -component, so that j =( j x − N y + j y ranges from 1 to N = N x N y . For theremainder of this paper, we adopt a two-index notation,where this is convenient, i.e. we may write z j x ,j y insteadof z j , denoting the x - and y -components by subscripts.Note that one has the freedom to rescale the coordi-nates since this changes the wave functions only by a totalfactor. The constant factor that we included in Eq. (19)is chosen such that the center of the cylinder is at theunit circle.We assume that the number of sites N is even. It isalso possible to study the case of N being odd which willshow the existence of two topological sectors. However,we can already identify the two anyonic sectors for N even. As will be shown below in Sec. III B, the state ψ and the singlet component ψ sgl0 of the state with two ad-ditional spins (one at z = 0 and one at z ∞ = ∞ ) can beobtained from the wave function of N primary fields onthe torus in the limit where the torus becomes a cylin-der. This argumentation shows that the two states ψ and ψ sgl0 represent the two anyonic sectors in the case ofan even number of spins. It would be possible to con-sider an odd number of sites on the cylinder by puttingan additional spin either at z = 0 or at z ∞ = ∞ so that the charge neutrality condition is satisfied. On the torus,however, such a construction is not possible and the ar-gumentation that we used to identify the two sectors foreven N does not directly apply. A. Global transformation properties of CFT states In this subsection, we study global transformationproperties of the states ψ , ψ a l ...a n l ...n , and ψ s ,s ∞ . Thisserves two purposes: First, it allows us to conclude thatstates with a different momentum are orthogonal, i.e.they have different global properties. Later, we will studytheir local behavior numerically and compare spin corre-lation functions in the bulk. The symmetries derived inthis subsection will be exploited in our numerical calcu-lations to obtain efficient Monte Carlo estimates.We consider the translation operator in the periodicaldirection T y and the inversion operator I . Their precisedefinition and the derivation of their action on the states ψ , ψ a l ...a n l ...n , and ψ s ,s ∞ are given in Appendix A. Geo-metrically, the translation operator rotates the system inthe periodical direction and the inversion operator corre-sponds to a reflection of the cylinder along its two centralcross sections. We call it an inversion because it acts onthe coordinates defined in Eq. (19) as z i → z − i .Eigenstates of T y and I are given in Table I. As weshow in Appendix A 3, applying the inversion I to ψ a l ...a n l ...n corresponds to inserting the current operators at z ∞ = ∞ instead of z = 0. We use the notation ψ a l ...a − n l ···− n forthese states: ψ a l ...a − n l ···− n ( s , . . . , s N ) ≡ (cid:104) J a n . . . J a l n l φ s ( z ) . . . φ s N ( z ) (cid:105) = ( u a l n l . . . u a n ψ )( s , . . . , s N ) . (20)Since the momentum in the periodical direction P y is re-lated to T y through the relation T y = e iP y , we concludefrom Table I that an additional insertion of a current op-erator J a − n into the correlation function of primary fieldsadds a momentum of − πn/N y to the state. In particu-lar, the states ψ and ψ a l ...a n l ...n have a different momentumif k = (cid:80) lj =1 n j is different from 0 modulo N y . B. Relation to the KL states on the torus In this subsection, we place the system on the torusand take a limit in which the torus becomes a cylinder.We show that the wave function of N primary fields onthe torus gives rise to ψ and the singlet component ψ sgl0 of ψ s ,s ∞ on the cylinder.We define the torus for ω > ω = iL with L > z with z + nω + mω for m, n ∈ Z . The two circumferences of the torus are there-fore given by ω and | ω | . Let us denote the positions onthe torus by v i , i.e. we assume that v i lie in the rectan-gle spanned by ω and ω . Keeping the positions v i fixed TABLE I. Eigenstates of the translation operator T y and theinversion I . The sum of mode numbers (cid:80) lj =1 n j is denotedby k . For the states ψ a l ...a − n l ···− n , the current operators areinserted at z ∞ = ∞ [cf. Eq. (20)].Eigenstate T y I ψ ( − N x N ( − N y N ψ a l ...a n l ...n ( − N x N e − πiNy k — ψ a l ...a n l ...n ± ψ a l ...a − n l ···− n — ( ± − N y N ψ s ,s ∞ : ( − N x N + N x — ψ ↑ , ↓ − ψ ↓ , ↑ ( − N x N + N x ( − N y N + N x ψ ↑ , ↑ , ψ ↑ , ↓ + ψ ↓ , ↑ , ( − N x N + N x ( − N y N + N x +1 and ψ ↓ , ↓ and taking the circumference L = | ω | → ∞ transformsthe torus into a cylinder, as illustrated in Fig. 1.0 ω = N y Re N x ω → i ∞ Im FIG. 1. (Color online) Limit in which the torus becomes acylinder: The circumference | ω | is taken to infinity while thepositions lie in the finite region of size N x × N y (red patch). On the torus, there are two states ψ torus k with k ∈{ , } . These are given by ψ torus k ( s , . . . , s N ) = (cid:104) φ s ( v ) . . . φ s N ( v N ) (cid:105) k ∝ δ s χ s θ (cid:34) k (cid:35) (cid:32) N (cid:88) i =1 ζ i s i , τ (cid:33) N (cid:89) i We calculated two-point spin correlation functions inthe states ψ , ψ a and in the singlet state ψ sgl0 ≡ ψ ↑ , ↓ − ψ ↓ , ↑ (27)using a Metropolis Monte Carlo algorithm. This allowedus to compare properties of the states numerically forlarge system sizes by sampling the relevant probabil-ity distributions. We furthermore exploited the transla-tion and inversion symmetries of Table I to average over0 5 10∆ y − | S zzψ | i x = j x = 70 5 10∆ y − | S zzψ | i x = j x = 1 − y j x i x = 7 − y j x i x = 1 − − − − − − − − − − − − − − − − − − − FIG. 2. (Color online) Two-point spin correlation function S zzψ ( i x , j x , ∆ y ) in the bulk (upper panels) and at the edge (lowerpanels) for N x = 13 and N y = 20. The left panels show the two-dimensional dependency in a color plot. Whenever thevalue of the correlation function does not differ from zero by more than three times the estimated error, we excluded the datapoint from the plot (blank fields). In the right panels, the absolute value | S zzψ ( i x , i x , ∆ y ) | of the correlation function along the y -direction is plotted. Points for which the sign of the correlation function is positive (negative) are shown in blue (red). Forthe data shown in gray, the mean value does not differ from zero by more that three times the estimated error. In the bulk,the correlations decay exponentially, while a nonzero, negative correlation remains at the edge for ∆ y ≥ equivalent correlation functions, thus obtaining a fasterconverging Monte Carlo estimate.In this subsection, we shall use the notation S abψ ( i x , j x , ∆ y ) = 4 (cid:104) ψ | t ai x , ∆ y +1 t bj x , | ψ (cid:105)(cid:104) ψ | ψ (cid:105) (28)for the two-point correlation function in a state ψ . Sinceall wave functions that we consider have a translationalsymmetry in the periodical direction, their value onlydepends on the difference ∆ y of the positions in the y -direction.Before comparing the wave functions with each other,we discuss the spin ordering pattern in ψ , which is en-coded in the correlation function S zzψ ( i x , j x , ∆ y ). (Since ψ is a singlet, xx -, yy - and zz -correlations are the sameand only correlation functions with a = b are nonzero.)Our numerical results are shown in Fig. 2. In the bulkof the system, we observe a ring-like structure with analternating magnetization. At the edge, the correlationsare anti-ferromagnetic at short distances. At larger dis-tances along the y -direction, however, the sign becomesstationary and a negative correlation remains. In thetwo-dimensional picture, the ordering is still character-ized by an alternating magnetization with the sign of thecorrelation function changing along the x -direction.We now discuss the question if the states ψ a can beconsidered as edge states. If so, then the local propertiesof ψ a and ψ in the bulk should be the same. Since these are encoded in the spin correlation functions, we com-pared the nearest-neighbor two-point correlators in thebulk for different system sizes. The relative differences (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S bcψ a ( i x , j x , ∆ y ) − S bcψ ( i x , j x , ∆ y ) S bcψ ( i x , j x , ∆ y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (29)are shown in Fig. 3 for a = b = c = z (left panels) and a = z, b = c = x (right panels). Correlation functions forother choices of a, b , and c either vanish or can be reducedto these due to the SU(2) invariance of ψ . In the upperpanels, the correlations along the y -direction are shownand in the panels of the lower two rows along the x -direction. We find that the relative differences approachzero exponentially as a function of N x . Even though thedifferences tend to be larger for smaller N y , they are stillexponentially suppressed as N x is increased. This is anindication that the wave functions ψ a indeed describeedge states compared to ψ as the thermodynamic limitin the open direction is taken.Our results for the comparison between ψ sgl0 and ψ areshown in Fig. 4. Since both ψ and ψ sgl0 are singlets, it isenough to compare the zz -correlations. Furthermore, thecorrelations in the positive and the negative x -directionare the same in the middle of the cylinder since ψ and ψ sgl0 are symmetric under the inversion, cf. Sec. III A.In contrast to ψ a , we find that the thermodynamic limitin the x -direction is not enough for the differences to − − N y = 4 N y = 8 N y = 12 N y = 1610 − − r e l a t i v e d i ff e r e n c e : ψ z t o ψ N x − − N x xy zz xy xxxy zz xy xxxy z z xy x x FIG. 3. (Color online) Comparison of nearest-neighbor bulkcorrelations in ψ z and ψ . The vertical axes show the relativedifferences | S bcψ z − S bcψ | / | S bcψ | with b = c = z (left panels)and b = c = x (right panels). Along the horizontal axis, thenumber of spins in the open direction ( N x ) is varied. Thedifferent curves correspond to configurations with different N y . The insets show for which sites the correlation functionswere computed. The sites in the central column of the insetscorrespond to the middle of the cylinder in the x -direction.The relative difference decreases exponentially in N x . vanish. Rather, we observe that the differences becomestationary if N y is held fixed and N x increased. As shownin the right panels of Fig. 4, the differences do, however,tend to zero exponentially as a function of N y if N x ischosen large enough. D. States at a higher level In the previous subsection, the states at level one incurrent operators were considered. We also comparedspin correlations in ψ an to those in ψ for higher valuesof n . For very large mode numbers n , only the terms atthe edge contribute to the sum in u a − n . To see this, let − − N x − − r e l a t i v e d i ff e r e n c e : ψ s g l t o ψ N y = 4 N y = 8 N y = 12 N y = 16 2 6 10 14 N y N x = 21 xy zz xy zz FIG. 4. (Color online) Comparison of nearest-neighbor bulkcorrelations in ψ sgl0 and ψ . The vertical axes show the relativedifferences | S zzψ sgl0 − S zzψ | / | S zzψ | . In the left panels, N x is variedalong the horizontal axes and the different curves correspondto different choices of N y . On the right panels, N y varies alongthe horizontal axes and N x is fixed. For N x large enough, thedifferences tend to zero exponentially as a function of N y . us consider u a − n − mN y = N (cid:88) j =1 z n + mN y j t aj (30) ∝ N x (cid:88) j x =1 e − πNy ( n + mN y ) j x N y (cid:88) j y =1 e − πiNy nj y t aj x ,j y . (31)For large values of m , the terms with j x > j x = 1. We denote the corresponding states with onecurrent operator by χ an : | χ an (cid:105) = lim m →∞ | ψ an + mN y (cid:105) ∝ N y (cid:88) j y =1 e − πi njyNy t a ,j y | ψ (cid:105) . (32)Fig. 5 shows the difference in nearest-neighbor corre-lations along the y -direction relative to ψ for N x = 13and N y = 8. The three curves correspond to the states ψ z , ψ z and χ z . As the position in the open direction is in-creased, the differences vanish exponentially for all threestates. We note that the differences are large at the leftedge ( i x = 1) and small at the right edge ( i x = 13). Thisagrees with the expectation that the operators u a − n arelocalized at the left edge. In contrast to the state ψ sgl0 ,the states ψ an perturb ψ only at one edge and their be-havior is therefore expected to approach that of ψ at theother edge. The results of Fig. 5 provide an indicationthat ψ an describe edge states also for n > − − i x − − r e l a t i v e d i ff e r e n c e : ψ t o ψ i x ψ = ψ z ψ = ψ z ψ = χ z xy zz xy xxxy z z xy x xN x = 13 , N y = 8 FIG. 5. (Color online) Relative difference | S abψ − S abψ | / | S abψ | innearest-neighbor correlators for ψ ∈ { ψ z , ψ z , χ z } [cf. Eq. (32)for the definition of χ a ]. The position i x in the x -direction isvaried along the horizontal axis. The plots in the left panelshave a = b = z and those in the right panels a = b = x . Inthe upper panels, the correlations along the y -direction areshown [ j x = i x , ∆ y = 1] and the lower panels correspond tocorrelations along the x -direction [ j x = i x + 1, ∆ y = 0]. We note, however, that the linear span of ψ an for n ∈{ , . . . , N − } contains not only edge states. The states ψ an can even be linearly combined so that ψ is perturbedat an arbitrary position j : t aj | ψ (cid:105) = N (cid:88) n =2 (cid:0) Z − (cid:1) jn | ψ an − (cid:105) , (33)where Z is the matrix defined in Eq. (15). This obser-vation can be understood from the fact that two states ψ am and ψ an are not necessarily orthogonal if m − n = 0modulo N y . In this case, ψ am and ψ an have the same mo-mentum in the y -direction, as discussed in Sec. III A. Thelinear combination | ψ an (cid:105) − e − π Nx − | ψ an + N y (cid:105) = N x (cid:88) j x =1 (1 − e − π ( j x − ) N y (cid:88) j y =1 z nj x ,j y t aj x ,j y | ψ (cid:105) , (34)for example, receives no contribution from spin operatorsat the left edge ( j x = 1). Even though both ψ an and ψ an + N y are perturbed from ψ mostly at j x = 1, this isnot the case for the difference of Eq. (34). E. Inner products of states from current operators In this subsection, we discuss the relation of innerproducts between the states ψ a l ...a n l ...n on the level of thespin system and CFT inner products between states( J a l − n l . . . J a − n )(0) | (cid:105) . For edge states in the continuumthat are constructed from descendant states of a CFT,the authors of Ref. 11 come to the remarkable conclusionthat, in the thermodynamic limit and under the assump-tion of exponentially decaying correlations in the bulk,the inner products between edge states are the same asthe inner products between CFT states. We now considerinner products between states constructed from currentoperators to test if a similar correspondence holds for thelattice states ψ a l ...a n l ...n and the CFT states they are con-structed from. The spin system inner products that weconsider are given by R k + k (cid:48) (cid:104) ψ a l ...a n l ...n | ψ b l (cid:48) ...b m l (cid:48) ...m (cid:105)(cid:104) ψ | ψ (cid:105)≡ R k + k (cid:48) (cid:104) ψ | (cid:0) u a − n (cid:1) † . . . (cid:0) u a l − n l (cid:1) † u b l (cid:48) − m (cid:48) l . . . u b − m | ψ (cid:105)(cid:104) ψ | ψ (cid:105) , (35)where N is the number of spins, R = min j ∈{ ,...,N } | z j | = e − πNy ( N x − (36)is the minimal absolute value of the positions, k = (cid:80) lj =1 n j , and k (cid:48) = (cid:80) l (cid:48) j =1 m j . The factor R k + k (cid:48) accountsfor the scaling of the operators u a − n with respect to arescaling of the positions. The minimal value is chosenbecause the operators u a − n = N (cid:88) j =1 t aj ( z j ) n (37)have the highest contribution at the edge with | z j | = R .We compare the inner products of the lattice sys-tem (35) to the CFT inner products (cid:104) | J a n . . . J a l n l J b l (cid:48) − m l (cid:48) . . . J b − m | (cid:105) . (38)If a correspondence similar to that of Ref. 11 also holdsfor lattice states, then the expressions of Eq. (35) shouldapproach those of Eq. (38) in the thermodynamic limit.Note that the inner products of the spin system arehard to evaluate for large system sizes, whereas the CFTinner product can be easily computed using the Kac-Moody algebra (2). On the level of the spin system, theinsertion of current operators corresponds to an applica-tion of spin operators to ψ [cf. Eq. (12)]. Therefore,the inner products can be determined numerically usinga Monte Carlo method if the number of current operatorsis small.We calculated the inner products for the states ψ a , ψ a and ψ b b , , which are all nonzero states at levels one andtwo. For these states, inner products between differentstates vanish because they have either a different spin ora different momentum. It is thus sufficient to comparethe norm squared of a state to the norm squared of thecorresponding CFT state, as summarized in the followingtable:Spin state CFT state Norm squared of CFT state ψ a J a − | (cid:105) (cid:104) J a J a − (cid:105) = (no sum over a ) ψ a J a − | (cid:105) (cid:104) J a J a − (cid:105) = 1 (no sum over a ) ψ b b , J b − J b − | (cid:105) (cid:104) J c J c J b − J b − (cid:105) = N y − r e l a t i v e d i ff e r e n c e N x = 11 ψ b b , ψ a ψ a 10 20 N y N x ψ a N y ψ a N y ψ b b , − − − FIG. 6. (Color online) Inner product of spin system states:Relative difference to the CFT expectation [cf. Eq. (39)].The colors correspond to the states ψ b b , (red), ψ a (blue) and ψ a (green). The upper panels show the relative differencein a color plot as a function of N x and N y . We observe avery weak dependence on N x for N x ≥ 3. The lower panelshows the dependence on N y for N x = 11. For large enough N y , the data are consistent with a power-law behavior withan exponent of approximately − . 1. Monte Carlo error barsare not plotted because they are barely visible on the chosenscale. The maximal relative error of the shown data points is0.31 %. In Fig. 6, our numerical results are shown for the rel-ative difference (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R k (cid:104) ψ | ψ (cid:105)(cid:104) ψ | ψ (cid:105) − (cid:104) ψ CFT | ψ CFT (cid:105)(cid:104) ψ CFT | ψ CFT (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (39)as a function of the system size. Here, ψ ∈ { ψ a , ψ a , ψ b b , } is one of the spin states, k = 1 for ψ = ψ a , k = 2 for ψ ∈ { ψ a , ψ b b , } , and ψ CFT is the CFT state correspondingto ψ . For a given system size, we observe a smaller differ-ence for the states at level k = 1 than for those at level k = 2. The computed inner products approach the CFTexpectation if N y is increased. The dependence on thenumber of spins in the x -direction is, however, very weakfor N x ≥ 3. In particular, the CFT result is not ap-proached if N x is increased and N y kept fixed. For largeenough N y , our data suggest that the spin system innerproducts approach the CFT result with a power law in N y . IV. LOCAL MODEL HAMILTONIAN In the previous section, we provided numerical evi-dence that the states with one current operator insertionrepresent edge states with respect to ψ . In this section,we study a set of local Hamiltonians on the cylinder. Fora suitable choice of parameters, the ground state of thecorresponding Hamiltonian has a good overlap with ψ and some of its low-energy excited states are well ap-proximated by ψ a , the states with one current operatorof order one.We study the local Hamiltonians H = J (cid:88) (cid:104) i,j (cid:105) t ai t aj + J (cid:48) (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) t ai t aj + J (cid:88) (cid:104) i,j,k (cid:105) (cid:9) ε abc t ai t bj t ck . (40)In these sums, the sites lie on a square lattice, (cid:104) i, j (cid:105) denotes all nearest neighbors, (cid:104)(cid:104) i, j (cid:105)(cid:105) all next-to-nearestneighbors, and (cid:104) i, j, k (cid:105) (cid:9) all triangles of nearest neighborsfor which i, j , and k are oriented counter-clockwise. Itwas shown in a previous study that the ground stateof H on the plane (open boundary conditions in both di-rections) and on the torus has a good overlap with theKL state for a range of parameters J , J (cid:48) , and J . Here,we study H on a cylinder of size N x × N y , where N x de-notes the number of sites in the open direction and N y the number of sites in the periodical direction. In thefollowing, we parametrize H in terms of two angles θ and θ : J = cos ( θ ) cos ( θ ) ,J (cid:48) = sin ( θ ) cos ( θ ) ,J = sin ( θ ) . (41)For N x = 5 and N y = 4, we studied the overlap be-tween ψ and the ground state ψ G of H as a function of θ and θ using an exact numerical diagonalization method.We also computed the overlap of the states with one cur-rent operator insertion at level one ψ a and the first ex-cited states ψ mE of H that have spin one and the samemomentum in the y -direction as ψ a . Here, m ∈ {− , , } denotes the T z eigenvalue of ψ mE .We denote the overlap between two states φ and φ as Ω ( φ , φ ) = |(cid:104) φ | φ (cid:105)|(cid:107) φ (cid:107)(cid:107) φ (cid:107) , (42)00 . . . θ / ( π ) |h ψ G | ψ i|k ψ G kk ψ k . 00 0 . 04 0 . 08 0 . 12 0 . θ / (2 π )0 . . . θ / ( π ) |h ψ E | ψ z i|k ψ E kk ψ z k . . . . . . FIG. 7. (Color online) Overlaps of states constructed fromCFT and eigenstates of the local Hamiltonian H of Eq. (40)for N x = 5 and N y = 4. The angles θ and θ parametrize thecoupling constants of H according to Eq (41). In the upperpanel, the overlap Ω ( ψ G , ψ ) ≡ |(cid:104) ψ G | ψ (cid:105)| / ( (cid:107) ψ G (cid:107)(cid:107) ψ (cid:107) ) be-tween ψ and the ground state ψ G of H is plotted. The lowerpanel shows the overlap between ψ z and the first excited state ψ E of H with the same spin and y -momentum as ψ z [spin one, T z = 0, momentum 3 / (8 π )]. The point marked with an opencircle has θ = 0 . × π and θ = 0 . × π and the highestcombined overlap of (cid:113) Ω ( ψ G , ψ ) + Ω ( ψ E , ψ z ) ≈ . where (cid:107) φ (cid:107) = (cid:112) (cid:104) φ | φ (cid:105) is the norm of a state. In Fig. 7,the overlaps Ω ( ψ G , ψ ) and Ω (cid:0) ψ G , ψ z (cid:1) are shown as afunction of the parameters of the Hamiltonian. Due toSU(2) invariance, it is sufficient to consider the overlapbetween the states ψ E and ψ z , which both have T z = 0: |(cid:104) ψ E | ψ +1 (cid:105)| = |(cid:104) ψ − E | ψ − (cid:105)| = |(cid:104) ψ E | ψ z (cid:105)| , (43)where ψ ± ≡ ψ x ± iψ y . The best value for the combinedoverlap (cid:113) Ω ( ψ G , ψ ) + Ω ( ψ E , ψ z ) was obtained for theangles θ = 0 . × π and θ = 0 . × π :Ω ( ψ G , ψ ) Ω (cid:0) ψ E , ψ z (cid:1) Ω ( ψ G , ψ ) N Ω (cid:0) ψ E , ψ z (cid:1) N . . . . N , the overlaps are expected to scaleexponentially in N . By taking the N th root, one obtainsa measure for the overlap per site, which takes into ac-count this exponential scaling. Notice that the overlapper site is higher than 99 % for both the ground and theexcited state.The low-energy spectrum of H for the parameters withthe best overlaps is plotted in Fig. 8. We find 8 en-ergies below the energy of ψ mE . The spectra plotted in 0 1 2 spin s E − E ∗ ψ : . p = 0 ψ z − : . p = 1 p = 2 ψ z : . p = 3 FIG. 8. (Color online) Low energy spectrum of the Hamilto-nian H of Eq. (40) for N x = 5 , N y = 4 , θ = 0 . × π , and θ = 0 . × π . The ground state energy of H is E ≈ − . y -momentum p/ (2 πN y ) with p ∈ { , , , } . Each shown level has a degen-eracy of 2 s + 1 corresponding to the values of T z . The labelsshow the ansatz state constructed from CFT and the valuefor its overlap with the corresponding eigenstate of H . The8 energies shown in red are those that are smaller than thelowest energy in the p = 1 and p = 3 sectors. [At the levelmarked with an asterisk ( ∗ ), there are actually two energieswith a splitting of approximately 1 . × − . This is notvisible on the scale of the plot.] Fig. 8 are separated into sectors of different y -momentum p/ (2 πN y ) = p/ (8 π ) with p ∈ { , , , } . Note that thestates ψ mE are the first excited states with p = 3. Thespectra for the momenta for p = 1 and p = 3 are thesame because H is invariant under the inversion opera-tor I introduced in Sec. III A: I − H I = H. (44)The relation I − T y I = T − y (45)between I and translation operator in the y -direction T y follows directly from their definition (cf. Appendix A 2and A 3). Therefore, if | ψ (cid:105) is an eigenstate of H with mo-mentum p/ (2 πN y ), then I| ψ (cid:105) is also an eigenstate withmomentum ( N y − p ) / (2 πN y ). This means that for | ψ mE (cid:105) with p = 3, there is a corresponding eigenstate I| ψ mE (cid:105) with p = 1, which satisfies |(cid:104) ψ E | ψ z (cid:105)| = |(cid:104)I ψ E | ψ z − (cid:105)| . (46)Here, | ψ z − (cid:105) = I| ψ z (cid:105) is the state obtained by insertingthe current operator at z ∞ = ∞ instead of z = 0 [cf.Eq. (20)].1Our results show that the state ψ and the states withone current operator of order one are good approxima-tions of low-energy eigenstates of H for N x = 5, N y = 4, θ = 0 . × π , and θ = 0 . × π . This raises thequestion if further eigenstates of H are effectively de-scribed by states constructed as CFT correlators. Wealso computed the overlaps of eigenstates of H with someadditional states constructed from current operators forhigher orders in current operators. At level two in cur-rent operators, the overlaps with the first excited statesthat have the same spin and momentum as our ansatzstates are given by 0 . ψ a (0 . . ψ b b , (0 . V. CONCLUSION This work studies trial wave functions for lattice FQHstates constructed as chiral correlators of the SU(2) WZW CFT. To each CFT state, characterized by a se-quence of current operators, we associated a correspond-ing state ψ a l ...a n l ...n of the lattice system. For continuoussystems, analogous states constructed from CFT wereproposed as FQH edge states previously . The factthat we work on the lattice allowed us to apply MonteCarlo techniques to test a central expectation for edgestates: That the local, bulk properties of different edgestates should be the same.For a system on the cylinder, we compared spin corre-lation functions in the states with one current operator( ψ a ) to the state with no current operators ( ψ ). Our nu-merical results show that the nearest-neighbor bulk cor-relations approach each other exponentially as the num-ber of spins in the open direction ( N x ) is increased. Onthe other hand, the states ψ a and ψ are different glob-ally since their spin and momentum are different. Thissupports the assumption that they describe edge states.We compared inner products of lattice states at levelsone and two in current operators to CFT inner productsof the corresponding descendant states. For large enough N y (periodical direction), the computed inner productsapproach the CFT expectation with a power law in N y .This suggests that there is a correspondence between in-ner products of states ψ a l ...a n l ...n and CFT inner productsin the thermodynamic limit. Such a correspondence wasfound for continuous wave functions in Ref. 11.Furthermore, we compared nearest-neighbor bulk cor-relations in ψ sgl0 to those in ψ , where ψ sgl0 is the singletcomponent of the state obtained by insertion of two ex-tra primary fields. In contrast to ψ a , we find that thecorrelations do not approach each other if the thermody- namic limit is taken only in the open direction. However,if N x is chosen large enough, the difference in correlationfunctions vanishes exponentially as a function of N y .We showed by an exact diagonalization that ψ has agood overlap with the ground state of a local Hamiltonianand ψ a with the first excited states that have the samespin and momentum as ψ a . This could be an indicationthat further low-energy excitations of that local Hamilto-nian are edge states described by the SU(2) WZW CFT.It would be interesting to investigate this relation in moredetail for larger system sizes and different topologies.We showed that the complete Hilbert space is coveredby the linear span of the states ψ a l ...a n l ...n and, therefore,only a subset of these states are edge states. For thestates with one current operator, we argued that not alllinear combinations of states ψ am describe edge modesbecause states with the same y -momentum can be non-orthogonal. It is possible to restrict the space of states toan orthogonal subset given by ψ am with m ∈ { , . . . , N y } .Taking the limit of large mode numbers could be an-other possibility of removing bulk states for the linearspan of ψ a l ...a n l ...n . More precisely, one can replace n i by n i + m i N y and then take m i → ∞ . In this limit, thesum in the operators u a − n i − m i N y only extends over theedge sites because all other positions are exponentiallysuppressed. The fact that this class of states (and alsolinear combinations of such states) is obtained from ψ by application of edge spin operators only, suggests thattheir complete span represents edge states. For one ofthese states, χ a , we did numerical tests that indeed indi-cated that χ a is an edge state. ACKNOWLEDGMENTS We acknowledge funding from the EU IntegratedProject SIQS, FIS2012-33642, the Comunidad deMadrid grant QUITEMAD+ S2013/ICE-2801 (CAM),the Severo Ochoa Program, and the Villum Foundation. Appendix A: Translation and inversion of states onthe cylinder1. Transformation under a permutation of the spins Both the translation operator T y and the inversion op-erator I act on a product state as a permutation of thespins. Such a permutation operator O τ is defined for thepermutation τ of N elements as O τ | s , . . . , s N (cid:105) = | s τ (1) , . . . , s τ ( N ) (cid:105) . (A1)The action of O τ on ψ and ψ s ,s ∞ can be rewrittenin terms of a permutation of the positions z i , which willfacilitate our calculations for T y and I . Our derivationof this transformation rule follows Ref. 19.2We consider the wave function˜ ψ z ,...,z N ( s , . . . , s N ) = δ s χ s N (cid:89) i 2. Translation in the periodical direction The translation operator T y is defined through the per-mutation ˜ T y :˜ T y ( i x , i y ) = (cid:40) ( i x , i y + 1) , if i y (cid:54) = N y , ( i x , , if i y = N y , (A8)where i x is the x -component and i y the y -component ofan index i .The signature of this permutation is given bysign( ˜ T y ) = ( − N x ( N y − = ( − N x , (A9)where we used that N = N y N x is even. In terms of thepositions, the transformation corresponds to a multipli-cation by a phase, z ˜ T y ( j ) = e πi/N y z j . Therefore,˜ ψ z ,...,z N ( s ˜ T − y (1) , . . . , s ˜ T − y ( N ) )= sign( ˜ T y ) ˜ ψ z ˜ T y (1) ,...,z ˜ T y ( N ) ( s , . . . , s N )= ( − N x δ s χ s N (cid:89) i 3. Inversion We require that the inversion I acts on the positionsdefined in Eq. (19) as z ˜ I ( i x ) , ˜ I ( i y ) = 1 z i x ,i y . (A14)This leads to the definition˜ I ( i x , i y ) = (cid:40) ( N x + 1 − i x , N y − i y ) , if i y (cid:54) = N y , ( N x + 1 − i x , N y ) , if i y = N y . (A15)We note that in our choice of z i , the center of the cylinderis at the unit circle. If this is not the case, then thedefinition of Eq. (A15) leads to an additional factor when˜ I is applied to z i .In order to determine the sign of the permutation, wearrange the state | s , . . . , s N (cid:105) in a matrix: | s , , . . . , s N x ,N y (cid:105) ∼ = s , . . . s ,N y s , . . . s ,N y ... ... ... s N x , . . . s N x ,N y . (A16)The transformed state is then given by I| s , , . . . , s N x ,N y (cid:105)∼ = s N x ,N y − s N x ,N y − . . . s N x , s N x ,N y s N x − ,N y − s N x − ,N y − . . . s N x − , s N x − ,N y ... ... ... ... ... s ,N y − s ,N y − . . . s , s ,N y . (A17)To bring the transformed matrix back to the originalform, we first reverse all N y columns and then reverse all N x rows excluding the last element of each row. A singlesequence of L elements can be reversed in L ( L − 1) steps.Therefore, the sign of the permutation is given bysign(˜ I ) = ( − N y N x ( N x − N x ( N y − N y − . (A18)We next determine the contribution from the coordi-nate part of the wave function ψ . Using Eq. (A14), we have˜ ψ z ˜ I (1) ,...,z ˜ I ( N ) ( s , . . . , s N )= ˜ ψ z ,...,z N ( s , . . . , s N ) N (cid:89) m As shown in Sec. IV, the edge states ψ a have a goodoverlap with low-lying excited states of a local model,for which ψ approximates the ground state. In this sec-tion, we analytically construct SU(2)-invariant, nonlocalparent Hamiltonians for some linear combinations of thestates ψ a l ...a n l ...n , i.e. Hamiltonians for which they are exacteigenstates with the lowest energy. 1. Construction of parent Hamiltonians The starting point of our construction is the operator C a = N (cid:88) i (cid:54) = j z i + z j z i − z j ( t aj + iε abc t bi t cj ) . (B1)In Appendix C, we explicitly compute the action of C a on states constructed from ψ by insertion of current op-erators, and show that C a does not mix the states ψ a l ...a n l ...n with different levels k = (cid:80) lj =1 n j if k < N y . This prop-erty is key to our construction of parent Hamiltonians:It allows us to treat the levels separately starting withthe lower levels, which have fewer states. The action of C a on states at level k is described by a matrix. For low k , the dimension of this matrix is considerably smallercompared to that of an operator acting on the completeHilbert space. Moreover, the size of the matrix dependsonly on the level k rather than the number of spins N .We next add a multiple of the total spin T a to C a anddefine the operators D an = C a + ( n + 1 − N ) T a , (B2)where n is an integer. The operator D an is also closed inthe subspace of states of level k if k < N y since T a doesnot mix states of different levels. For certain values of n , we managed to find states constructed from currentoperators that are annihilated by the three operators D an for a ∈ { x, y, z } . These states are then ground states ofthe Hamiltonian H n = ( D an ) † D an , (B3)where the index a is summed over. Note that the Hamil-tonian H n is positive semi-definite and SU(2) invariant. H n is nonlocal and contains terms with up to four-bodyinteractions since D an has terms linear and quadratic inspin operators. Before describing our results, we note that the condi-tion D an | ψ (cid:105) = 0 for all a implies that the state ψ is part ofthe subspace on which T b T b and D an commute. To showthis, we first note that (cid:2) D an , T b (cid:3) = iε abc D cn , (B4)which is a direct consequence of the definitions ofEqs. (B1) and (B2). It then follows that (cid:2) T b T b , D an (cid:3) | ψ (cid:105) = (cid:0) − iε bac T b D cn − iε bac D cn T b (cid:1) | ψ (cid:105) = − iε bac (cid:2) D cn , T b (cid:3) | ψ (cid:105) = ε bac ε cbd D dn | ψ (cid:105) = 0 , (B5)where we assumed that D an | ψ (cid:105) = 0 for all a . The statessatisfying D an | ψ (cid:105) = 0 can therefore be decomposed intosectors of different total spin.We note that the condition (cid:2) T b T b , D an (cid:3) | ψ (cid:105) = 0 isequivalent to (cid:2) T b T b , C a + (1 − N ) T a (cid:3) | ψ (cid:105) = 0 , (B6)since T b T b and T a commute. The operator C a + (1 − N ) T a has the advantage that its matrix entries in termsof the states at level k do not depend N and n [cf.Eq. (C8) in Appendix C ]. In our calculations, we found ittechnically easier to first determine the subspace of stateson which T b T b and C a + (1 − N ) T a commute and thenlook for states that are annihilated by D an for a suitable n within that subspace. TABLE II. States constructed from current operators thatare annihilated by D an for a ∈ { x, y, z } on a cylinder with N y > k [cf. Eqs. (B1) and (B2) for the definition of D an ]. For N y sufficiently large ( N y > k ), these states are ground statesof the Hamiltonian H n = ( D an ) † D an . k State Spin n ψ ψ a ψ a + iε ade ψ d e , ψ a b , ψ a d d , , + iε ade ψ d e , + iε ade ψ d e , + ψ a ψ a d d , , + 4 ψ d a d , , + ψ a d d , , + ψ a d d , , + iε ade ψ d e , iε ade ψ d e , + iε ade ψ d e , + ψ a iε ade ψ b d e , , + iε ade ψ b d e , , − ψ a b , − ψ a b , − ψ a b , ψ a b c , , iε ade ψ d e f f , , , − ψ a d d , , − ψ a d d , , − ψ a d d , , ψ d d a , , − ψ d a d , , − ψ a d d , , − ψ d a d , , − ψ a d d , , − ψ a d d , , − iε ade ψ d e , − iε ade ψ d e , − iε ade ψ d e , − iε ade ψ d e , − ψ a We summarize our analytical results in Table II. Thestates with spin 2 and 3 appear as the symmetric-5traceless parts of states with 2 and 3 open indices, respec-tively. For a two-index state φ ab , the symmetric-tracelesspart is defined as3( φ ab + φ ba ) − δ ab φ dd , (B7)and for a three-index state φ abc as (cid:0) φ abc + φ bca + φ cab + φ cba + φ bac + φ acb (cid:1) − (cid:0) δ ab ( φ cdd + φ dcd + φ ddc ) + δ ac ( φ bdd + φ dbd + φ ddb )+ δ bc ( φ add + φ dad + φ dda ) (cid:1) . (B8)Except for the levels 2 and 6, we find states and cor-responding parent Hamiltonians for all levels that wereconsidered. Note that the singlet ψ is a ground state of H n for any value of n . For the additional ground states,we observe that the value of n tends to be larger at higherlevels k . This means that the ground state space of theHamiltonians H n with lower n contains states of a lowerlevel in current operators. For example, we only find theground states ψ and ψ a for H . Similarly, the only ap-pearing ground states of H at levels k ≤ ψ andthe symmetric-traceless part of ψ a b , . 2. Ground-state degeneracies In the previous subsection, we explicitly constructedanalytical ground states of the Hamiltonians H n with n ∈ { , , , , , , } in terms of linear combinationsof states ψ a l ...a n l ...n with levels k ≤ 9. We now study theground state spaces of the Hamiltonians H n numericallyand provide evidence for n ∈ { , , } that the completeground state space is spanned by the states given in Ta-ble II. TABLE III. Numerically determined ground state multipletsof the Hamiltonians H n for n ≤ 13 and an even number ofspins N with N ≤ 14. The second column indicates the mini-mal number of spins N min y in the periodical direction for whichthe complete shown multiplet was observed in all system with N min y ≤ N y ≤ 14. For a lower number of spins in the y -direction, the observed ground state space is smaller. Foreven n , we only find a singlet ground state. n N min y Ground state multiplet1 2 0 ⊕ 13 4 0 ⊕ 25 6 0 ⊕ ⊕ 37 8 0 ⊕ ⊕ 49 10 0 ⊕ ⊕ ⊕ 511 12 0 ⊕ ⊕ ⊕ 613 14 0 ⊕ ⊕ ⊕ ⊕ By an exact diagonalization, we numerically deter-mined the ground state multiplets of the Hamiltonians H n for n ≤ 13 and systems with N = N x N y ≤ 14 and N even. Our results are summarized in Table III. Weobserve that states with spin s occur in the ground statespaces only in systems with N y ≥ s . Furthermore, wefind that the ground state degeneracy does not grow any-more if N y reaches a certain value N min y . This statementis most conclusive for the lower values n , where N min y is smaller and we are thus able to probe more systemswith N y ≥ N min y . For n ∈ { , , } , this implies that allground states are given by the corresponding states ofTable II.Finally, let us formulate a conjecture about the struc-ture of the states annihilated by D an , which are groundstates of H n . Our analytical results are consistent withthe following rule: For each spin sector s ∈ { , , . . . } ,there is a series of states at levels k = s + 2 sj with j ∈ { , , , . . . } . These states are annihilated by D an with n = 2 s − j . As one can show by induction, thesecond rule implies that the ground state space of H n with n = 2 s − ⊕ (cid:40) ⊕ ⊕ · · · ⊕ s, if s is odd , ⊕ ⊕ · · · ⊕ s, if s is even.The numerical results of Table III are consistent with thismultiplet structure and thus support the conjecture thatthe values of n are given by n = 2 s − j . Appendix C: Action of C a on states built fromcurrent operators Our starting point is the decoupling equation for thestates ψ a k ...a ... derived in Ref. 33. This equation describesthe action of the operator C ai = (cid:88) j ∈{ ,...,N }\{ i } z i + z j z i − z j ( t aj + iε abc t bi t cj ) (C1)on the states ψ a k ...a ... and follows from the CFT null field( K ab ) i ( J b − φ s i )( z i ) with ( K ab ) i = δ ab − iε abc t ci . (C2)[The definition of C ai used here differs from that of Ref. 33by a factor of 2 / C a = N (cid:88) i =1 C ai (C3)was used in Sec. B to construct parent Hamiltonians for states built from current operators.The decoupling equation reads C ai | ψ a k ...a ... (cid:105) = k (cid:88) q =1 ( K aa q ) i z i | ψ a k ...a q +1 a q − ...a ... ... (cid:105) + ( K ab ) i T b | ψ a k ...a ... (cid:105) + 2( K ab ) i (cid:88) s ,...,s N k (cid:88) q =2 q − (cid:88) n =0 iε ba q c z n +1 i (cid:104) Φ s ( z )( J a k − . . . J a q +1 − J cn J a q − − . . . J a − )(0) (cid:105)| s , . . . , s N (cid:105) , (C4)where Φ s ( z ) = φ s ( z ) . . . φ s N ( z N ) . (C5)The decoupling equation for the states ψ a k ...a ... , where all mode numbers are one, is enough to describe the action of C ai on states with general mode numbers ψ a l ...a n l ...n : Using the Kac-Moody algebra of Eq. (2), the latter can be expressedin terms of the states ψ a k ...a ... by repeated application of J a − n = i ε abc (cid:2) J c − , J b − n +1 (cid:3) ( n (cid:54) = 0) . (C6)On the cylinder, we have N (cid:88) i =1 ( z i ) − n = 0 , if n mod N y (cid:54) = 0 . (C7)Summing over i in Eq. (C4), we therefore obtain for k < N y C a | ψ a k ...a ... (cid:105) = ( N − T a | ψ a k ...a ... (cid:105) + k (cid:88) q =1 iε a q ac | ψ ca k ...a q +1 a q − ...a ... ... (cid:105) + (cid:88) s ,...,s N k (cid:88) q =2 q − (cid:88) n =0 G q,na k ...a ( s , . . . , s N ) | s , . . . , s N (cid:105) , (C8)with G q,na k ...a ( s , . . . , s N ) = 2 (cid:104) Φ s ( z )( J a q − n − J a k − . . . J a q +1 − J an J a q − − . . . J a − )(0) (cid:105)− δ a q a (cid:104) Φ s ( z )( J c − n − J a k − . . . J a q +1 − J cn J a q − − . . . J a − )(0) (cid:105) . (C9)The first two terms on the right hand side of Eq. (C8) are of order k in current operators since T a | ψ a k ...a ... 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